Ostrowski-type inequalities for fuzzy-valued functions and its applications in quadrature theory
It is a difficulty that fuzzy spaces cannot be equipped with a vectorial structure, i.e., it is not possible to use in a direct way tools developed on classical functional analysis. This is also the case in differential and integral calculus theory for fuzzy sets-valued functions. The first concept of differentiability for fuzzy-valued functions was presented by \textit{M. L. Puri} and \textit{D. A. Ralescu} [J. Math. Anal. Appl. 91, 552--558 (1983; Zbl 0528.54009)], which however is very restrictive. Today the generalized Hukuhara differentiability (gH) is used in differential and integral calculus for fuzzy-valued functions. In this context, the concept of switching points is fundamental since the gH-derivative of a fuzzy-valued function at a point can be completely determined when it is known if such a point is or is not a switching point. Many results as the fundamental theorem of calculus and the mean value theorem are introduced under assumption of the existence of a finite number of switching points. In this paper, the authors show that the set of switching points is a countable set. Using this fact, fuzzy versions of the fundamental theorem of calculus, of the mean value theorem, and of the Ostrowsky inequality are presented. Also, new quadratur rules are introduced and discussed.
scientific article; zbMATH DE number 7341968
scientific article